Optimizing Multivariate Function with Lagrange Multiplier Method

[hawaiifiver]

Homework Statement

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

Homework Equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

The Attempt at a Solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Homework Statement

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

Homework Equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

Okay, that's good. What I would do is switch all of those "\lambda" terms to the right:
\frac{-m c^m}{u^{m+1}}= 2\lambda u
\frac{-m d^m}{v^{m+1}}= 2\lambda v
\frac{-m d^m}{w^{m+1}}= 2\lambda w

(In fact, to find extrema of f with constraint g= constant, I tend to think of the Lagrange multiplier condition as \nabla f= \lambda \nabla g rather than using the "auxiliary function" f+ \lambda g. It gives the same result, of course.)

Now, since a particular value of \lambda is not part of the solution start by eliminating \lambda by dividing one equation by another. Dividing the first equation by the second, for example, gives
\left(\frac{c}{d}\right)^m\left(\frac{v}{u}\right)^{m+1}= \frac{u}{v}
which gives
\left(\frac{v}{u}\right)^{m+2}= \left(\frac{d}{c}\right)^m
and similarly for the other equations.

The Attempt at a Solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

 

[Ray Vickson]

Homework Statement

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

Homework Equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

The Attempt at a Solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

The Attempt at a Solution

You have expressions for u^{m + 2} , v^{m+2} , w^{m+2} in terms of λ. You can take (m+2) roots, to get u, v, w. You can then substitute these expressions into the constraint, to get a single, simple equation for λ. Actually, it is easier to re-write everything in terms of μ = -λ; that will eliminate a lot of minus signs in the equations and solutions.

RGV

 

[hawaiifiver]

Thank you both. I'm understanding this problem now.